Prove that `sum_(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10)dot`

If cos alpha + cos beta + cos gamma = 0 and also sin alpha + sin beta + sin gamma= 0, then prove that: cos 3 alpha + cos 3beta + cos 3gamma = 3 cos (alpha + beta + gamma)

If alpha,beta,gamma, in (0,pi/2) , then prove that (s i(alpha+beta+gamma))/(sinalpha+sinbeta+singamma)<1

Let x-ysin alpha-zsin beta=0 , xsin alpha+zsin gamma-y=0 and xsin beta+ysin gamma-z=0 be the equations of the planes such that alpha+beta+gamma=pi//2 (where alpha,beta and gamma!=0)dot Then show that there is a common line of intersection of the three given planes.

Prove that: sinalpha+sinbeta+singamma-sin(alpha+beta+gamma)=4sin((alpha+beta)/2)sin((beta+gamma)/2)sin((gamma+alpha)/2)dot

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+alpha)/2)

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+alpha)/2)

Prove that: cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+alpha)/2) .

Prove that |2alpha+beta+gamma+deltaalphabeta+gammadelta alpha+beta+gamma+delta2(alpha+beta)(gamma+delta)alphabeta(gamma+delta)+gammadelta(alpha+beta) alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

If cosalpha+cosbeta+cosgamma=0 a n d a l so sinalpha+sinbeta+singamma=0, then prove that cos2alpha+cos2beta+cos2gamma =sin2alpha+sin2beta+sin2gamma=0 sin3alpha+sin3beta+sin3gamma=3sin(alpha+beta+gamma) cos3alpha+cos3beta+cos3gamma=3cos(alpha+beta+gamma)

If alpha,beta,gamma are different from 1 and are the roots of a x^3+b x^2+c x+d=0a n d(beta-gamma)(gamma-alpha)(alpha-beta)=(25)/2 , then prove that |alpha/(1-alpha)beta/(1-beta)gamma/(1-gamma)alphabetagammaalpha^2beta^2gamma^2|=(25 d)/(2(a+b+c+d))